Answer:
[tex] \frac{1}{9} [/tex]
Step-by-step explanation:
Let the numbers be x,y, where x>y
The geometric mean is
[tex] \sqrt{xy} [/tex]
The Arithmetic mean is
[tex] \frac{x + y}{2} [/tex]
The ratio of the geometric mean and arithmetic mean of two numbers is 3:5.
[tex] \frac{ \sqrt{xy} }{ \frac{x + y}{2} } = \frac{3}{5} [/tex]
We can write the equation;
[tex] \sqrt{xy} = 3 [/tex]
or
[tex]xy = 9 - - - (2)[/tex]
l
and
[tex] \frac{x + y}{2} = 5[/tex]
or
[tex]x + y = 10 - - - (2)[/tex]
Make y the subject in equation 2
[tex]y = 10 - x - - - (3)[/tex]
Put equation 3 in 1
[tex]x(10 - x) = 9[/tex]
[tex]10x - {x}^{2} = 9[/tex]
[tex] {x}^{2} - 10x + 9 = 0[/tex]
[tex](x - 9)(x - 1) = 0[/tex]
[tex]x =1 \: or \: 9[/tex]
When x=1, y=10-1=9
When x=9, y=10-9=1
Therefore x=9, and y=1
The ratio of the smaller number to the larger number is
[tex] \frac{1}{9} [/tex]
